Constant AccelerationConstant Acceleration

Acceleration is how quickly the velocity is changing - the change in velocity per unit time. If the acceleration is constant, this can then be expressed as the following equation:

Where:

a is acceleration

v is the final velocity

u is the initial velocity

t is time taken

Acceleration = Change in VelocityTime

a = (v - u)t

Change in velocity = Final velocity - Initial velocity

at + u = v (add u to both sides)

Constant AccelerationConstant Acceleration

From...

a = (v - u)t

(multiply both sides by t)at = v - u

Rewriting, gives...

We rearrange the formula to make v the subject...

v = u + at

Calculating the Distance TravelledCalculating the Distance Travelled

Consider an object accelerating from an initial velocity u to a final velocity v in t seconds...

As before, to calculate the distance travelled, we need to find the area underneath the graph...

Area of Blue Triangle= ½ Base x Height= ½ x t x (v - u)= ½ x t x (at)= ½ at2

Calculating the Distance TravelledCalculating the Distance Travelled

Total Area = ut + ½ at2 so theformula for distance travelled (s), is…

Area of Lilac Rectangle= Base x Height= t x u= ut

s = ut + ½ at2

tat = v - u

a = (v - u)From:

If we multiply both sides of the equation by t we can find out s...

We also know that the average speed can be calculated from...

This formula allows us to calculate the distance travelled if we know the initial and final velocities and the time taken.

Average speed = Total distance travelledTotal time taken

u + v = st2

Average SpeedAverage Speed

s = (u + v) x t2

Constant Acceleration FormulasConstant Acceleration Formulas

We now have 3 constant acceleration formulas…

v = u + at

v - u = at

v - u = ta

t = v - ua

(dividing both sides by a)

(subtracting u from both sides)

(rewriting)

s = (u + v) x t2

s = ut + ½ at2v = u + at

We can combine formulas and by eliminating t...

The first step is to make t the subject of formula

s = (u + v) x t2

s = (u + v) x2

(v - u)a

2as = (u + v) (v - u)

2as = v2 - u2

v2 = u2 + 2as

Multiply both sides by 2a...

Multiply out the brackets...

Add u2 to both sides and rewrite...

Constant Acceleration FormulasConstant Acceleration Formulas

So the fourth constant acceleration formula is:

v2 = u2 + 2as

t = v - ua

The second step is to substitute this value for t into formula

Constant Acceleration Formulas - SummaryConstant Acceleration Formulas - Summary

Symbol

Meaning Unit

v final velocity m/s

u initial velocity m/s

s distance travelled

m

t time taken s

a acceleration m/s2

We now have 4 formulas. These formulas will help you calculate any motion problem in which a body undergoes zero or constant acceleration.

Formulas

s = (u + v ) x t 2

s = ut + ½at2

v = u + at

v2 = u2 + 2as

Whenever you have any 3 of the five ‘v-u-s-t-a’ unknowns, you can find out the remaining 2 unknown values by using one or more of the above formulas…

Using Constant Acceleration FormulasUsing Constant Acceleration Formulas

v = u + atv = u + at

s = ut + ½at2 s = ut + ½at2

v2 = u2 + 2asv2 = u2 + 2as

s = (u + v ) x t s = (u + v ) x t22

Horizontal

Vertical

Ignoring air resistance, the only force acting on a projectile during the flight is gravity.

Projectiles have a downward acceleration (due to gravity) and this only affects the vertical velocity.

For a projectile there is no acceleration in the horizontal direction.

Projectile Motion - Forces ActingProjectile Motion - Forces Acting

Altering Projectile VelocityAltering Projectile VelocityA cannon ball is fired horizontally at a target. Can you find out (or calculate) the correct velocity to hit the target?

Calculating Time TakenCalculating Time TakenExample: Calculate the time taken, from firing, for the cannon ball to hit the target.

t = d/s

t = 48/24

Time taken is 2s

t = 2

t = d/s is a formula that can be applied to solve problems, whenever velocity is constant...

Velocity is constant in the horizontal vector...

Calculate the final vertical velocity of the cannon ball as it hits the target. In this case a = g = 9.8ms-2 (9.8m/s2)

Calculating Final VelocityCalculating Final Velocity

Example:

v = u + at

v = 0 + 9.8 x 2

v = 19.6m/s

v = u + at is a formula that can be applied to solve problems, whenever acceleration is constant...

Final vertical velocity is 19.6m/s

v = 0 + 19.6

Kinetic Energy

2 starting equations Force = m x a (Newton’s 2nd law) W = Force x distance Together give

Work = (m x a) x distance

W = m x a x d

Work is MAD

Av Speed (velocity) is distance

time

Distance = Average velocity x time

Distance = v + u x t

2

Distance

Distance = v + u x t

2

D = ½ x v + u x t

Starting velocity is zero

D = ½ x v x t

Time

Final velocity = acceleration x time

Time = final velocity

acceleration

t = v

a

Going back a step…..

D = ½ x v x t

D = ½ x v x v

a

D = ½ x v2

a

Put into previous statements

W = m x a x d

W = m x a x ½ x v2

a

W = m x ½ x v2

W = ½ x m x v2

GPE work done = Force x distance

Force = mass x acceleration

Gravitational constant (g) is the cause of the acceleration

Force = mass x gravitational constant

F = m x g

Work = mass x acceleration x distance

In this case height is the distance

therefore, PE=mgh

Work

Work = force x distance

Measured in Joules. Weight is a force, mass is not

Power

Power = work / time (s)

Measured in Joules/second or Watts

Newton’s 2nd law

Force = mass x acceleration

Force in Newton’s Mass in kg Acceleration in m/s2